Strategy #1: Avoid Flow. Do What Does Not Come Easy.
“The mistake most weak pianists make is playing, not practicing. If you walk into a music hall at a local university, you’ll hear people ‘playing’ by running through their pieces. This is a huge mistake. Strong pianists drill the most difficult parts of their music, rarely, if ever playing through their pieces in entirety.”

Strategy #2: To Master a Skill, Master Something Harder.
“Strong pianists find clever ways to ‘complicate’ the difficult parts of their music. If we have problem playing something with clarity, we complicate by playing the passage with alternating accent patterns. If we have problems with speed, we confound the rhythms.”

Strategy #3: Systematically Eliminate Weakness.
“Strong pianists know our weaknesses and use them to create strength. I have sharp ears, but I am not as in touch with the physical component of piano playing. So, I practice on a mute keyboard.”

Strategy #4: Create Beauty, Don’t Avoid Ugliness.
“Weak pianists make music a reactive task, not a creative task. They start, and react to their performance, fixing problems as they go along. Strong pianists, on the other hand, have an image of what a perfect performance should be like that includes all of the relevant senses. Before we sit down, we know what the piece needs to feel, sound, and even look like in excruciating detail. In performance, weak pianists try to reactively move away from mistakes, while strong pianists move towards a perfect mental image.”

Of course as we are talking about studying for a public performance of music these strategies don’t translate immediately or perfectly to the study of mathematics. Nonetheless, I’ll give it a go.

1. Avoid Flow. Do What Does Not Come Easy.
The second part is crucial and is where many students make a big mistake. They get good at half of the course and hope that that will get them through the exam. I saw it in contemporaries at university and I see it in some of my students. They do they easy parts, for example focussing on doing the calculations rather than the concepts.

Hence the advice is to do the deeper work. I.e., become competent in not just the superficial calculations but have deeper understanding. For example, don’t just know what the definition is, know why it is the way it is and — hardest of all — know why it is not some other way.

2. To Master a Skill, Master Something Harder.
When I was first studying group theory I went further by studying why equations of degree 5 do not have solutions via radicals – a topic which was not part of the course. I found that doing this not only helped me grasp the basics of the material such as the definition of simple group but also helped with the harder stuff such as Sylow p-groups.

Mathematics is a linear subject in the sense that one concept is built upon another. Despite this we don’t have to — and probably don’t — learn it in a linear way. When learning proofs I would often try to learn the proofs from the end of the course first. One, this told me what was important in the course — if Lemma 2.1 kept coming in the later parts of the course, then it was a good idea to pay attention to it. Two, when it came to study the earlier parts in depth these seemed much easier. (An important point: you have to at least have some idea of what is in the start of the course, you can’t start in the middle, what I am saying is you don’t necessarily need to learn the start in depth to begin with.)

There is another good reason for studying the last half of the course. The harder parts of a course are often there because they are needed as prerequisites for later courses.

3. Systematically Eliminate Weakness.
When studying vector calculus I found that my weak point was calculating accurately the various types of integrals – I could quote the definitions and theorems such as Stokes’ Theorem without problem – but I kept messing up the calculations by forgetting terms, eg, missing the side of a cube. So I got Schaum’s outline series book on vector calculus and did many examples from there until I felt my accuracy had improved.

4. Create Beauty, Don’t Avoid Ugliness.Hardy famously said there is no room for ugliness in mathematics (I rather less famously said, “Then why do they write so many ugly papers”). But how to make point 4 relevant to mathematics? Well, although it can be seen above that I took action to avoid mistakes (ugliness), my main ambition was to have an overall understanding of the course so that everything fitted together perfectly for me. One method I used to achieve this was to rewrite the lecturer’s notes. I don’t mean word-for-word rewriting but trying to make major improvements. Lecturers rarely provide perfect notes (you try writing over thirty hours of material) and many theorems can be generalized with only a little bit of effort. I sometimes found that when I had a generalized theorem, then I could deduce some of the theorems from the course with ease.

For me, that was where the beauty was! Making it simple.

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